Tutorial elaborations part 2 - Microeconomics 2: Welfare economics (30L106-B6)
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Course
Microeconomics 2: welfare economics (30L106B6)
Institution
Tilburg University (UVT)
This document contains all the tutorial effects of part 2, the course Microeconomics 2: welfare economics given at Tilburg University. This section contains all material from after the midterm, because these parts are tested separately. These effects were written using an iPad with clear handwritin...
MICROECONOMICS 2: WELFARE ECONOMICS
Topic: Fairness, Poverty and Inequality
Exercise 1: Fairness
Consider the situation in which a cake of size 1 has to be divided between two persons. We assume
that each person ! is selfish and has a utility function "! that is strictly increasing in the part of the cake
#! that he/she gets. Hence, more is better. Assume that you want to make a fair division, and that your
fairness criteria is that each person should not feel envious of what the other person has (let’s call this
envy-free criteria). In other words, after allocating the cake each person should weakly prefer his own
part to the part that the other got: "" (#" ) ≥ "" (## ) and "# (## ) ≥ "# (#" ).
a) Is the situation in which each person gets one-third of the cake and one-third is thrown away
envy-free? Is it also Pareto efficient?
b) Which divisions of the cake are both Pareto efficient and envy-free?
c) Now assume that the cake has a rectangular shape. Furthermore, the cake has a nice cherry
on the left hand side, while there is no cherry on the right hand side. Person 1 (from now
on (" ) does not care about the cherry (it leaves him indifferent), but Person 2, (from now on
(# ) is willing to give up one-fourth of the cake to get the cherry. Both individuals know this.
Which divisions of the cake are both Pareto efficient and envy- free?
d) Assume (" and (# are using the “divide and choose” method to divide the cake. (" divides the
cake into two pieces and then (# can choose which part he takes; the part that is not taken is
then left for (" . How will (" divide the cake? Is the outcome Pareto-efficient? Is it envy-free?
e) Now consider the case from part d), but assume that the roles of the individuals are reversed:
(# divides and (" chooses. What is the outcome in this case? Is the outcome Pareto efficient?
Is the outcome envy-free?
f) Think about the envy-free criteria: what assumptions are needed to make it implementable?
g) Assume that a society adopts envy-free as its fairness criteria: after all, a society where nobody
is envious could be a happy society. Discuss what could be the resulting income distribution
in such a society.
Exercise 2: The Gini Index before and after taxation
Consider a society in which there is a continuum of individuals. Specifically, the society is represented
by the interval [0,1] and we denote a typical individual from this society by *. The total size of the
society is normalized to 1 and the gross income of individual *+[0,1] is 10*.
a) Show that the total income in this society is 5.
1
,b) For #+[0,1], let 1(#) be the share in the total income that is earned by all *+[0, #] together.
Hence, 1(. ) Is the Lorenz curve. Calculate 1(#) for each value of #.
c) Calculate the Gini-index of the gross income distribution.
d) Now assume that proportional taxation is introduced. Hence, each individual * has to pay a
fraction 2 from his income, so that his net income is (1 − 2)10*. What is the Gini index of the
net income distribution?
2
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